Isomorphism between oscillator and Coulomb-like theories in one and two dimensions

We give a mathematically rigorous description of dual nonrelativistic quantum oscillators and Coulomb-like theories in one and two dimensions. We compare their spectra and the full systems of (generalized) eigenfunctions of self-adjoint Hamiltonians. We consider all self-adjoint Schrödinger operators of these theories and present rigorous solutions of the spectral problem. To construct self-adjoint extensions, we use the method of (asymptotic) self-adjoint boundary conditions. In solving the spectral problem, we use Krein's method of guiding functionals. We show that there exists an isomorphism of the spectra of dual theories (in both the discrete and the continuous parts of the spectra) and an isomorphism of the corresponding full systems of (generalized) eigenfunctions of the self-adjoint Hamiltonians.